Modified Entropy Correction Formula for the Roe Scheme

To avoid un-realistic solutions like expansion shocks from appearing as a part of a solution it is necessary to satisfy the entropy condition for the Roe scheme. A variety of entropy fix formulae for the Roe scheme have been addressed in the literature. Three of the most famous are due to Harten-Hyman and Hoffmann-Chiang. These formulations have been assessed in this paper by applying them to the inviscid Burgers’ equation and shock tube problem. These entropy fix formulations are unable to totally diminish the expansion shock in the vicinity of sonic expansions. Moreover, they are not universal, i.e. a single formulation is not adequate for the scalar Burgers’ equation, shock tube problem and multi-dimensional cases and different formulations were suggested for each case. A simple modification to the Harten formulation is presented in this paper. This modification basically enlarges the band over which the entropy fix condition is enforced. The resulting formulation is able to totally remove the non-physical expansion shocks from the region of sonic expansion without affecting the rest of the computational domain. Comparison among the exact solution, and the entropy correction formulae of HartenHyman and Hoffmann-Chiang and the currently modified formula are shown here. The modified entropy fix formulation can totally diffuse the expansion shock. Moreover, the current formula does not affect the solution in the rest of the computational domain. Besides the modified formula, as a single formulation, can universally be applied over a wide range of applications from scalar equations to the governing equation of fluid motion. Finally in this paper, the following test cases are performed to assess the accuracy of the modified entropy formulation: inviscid shear flow, transonic flow over a bump, and transonic flow in a Laval Nozzle. Very accurate results are obtained. In the current study a second order upwind scheme of Roe with minmod flux limiter is applied. Ph.D. Candidate, AIAA student member. Professor, Associate Fellow AIAA. Copyright c ©2001 by M. J. Kermani & E. Plett. Published by the American Institute of Aeronautics and Astronautics, Inc. with permission. Introduction A scalar Riemman problem, such as the inviscid Burgers’ equation subject to an initial condition with discontinuity, could easily be solved by an Exact Riemman Solver (ERS) like the Godunov method. However, when an ERS is applied to the governing equations of fluid motion, it will give a computationally inefficient iterative technique. For example, consider the governing equation of unsteady fluid motion for a one-dimensional flow, like the shock tube problem. This equation can be expressed in a fully conservative form as follows: ∂Q ∂t + ∂F ∂x = 0 (1) where t is time, x is space, Q is the solution vector, and F is the flux vector. This equation introduces a non-linear set of three coupled scalar equations for the one-dimensional case. Solving these equations by any type of iterative exact method, such as the ERS, is not computationally efficient. One alternative idea that has been successfully used is to employ Approximate Riemman Solvers (ARS) for the fluid equations. One of the most popular ARS, has been proposed by Roe (1981). The ARS of Philip Roe (1981) suggests solving the linearized form of Eqn. 1, i.e. ∂Q ∂t + Ā ∂Q ∂x = 0 (2) where according to Roe (1981), Ā is a locally constant matrix and is determined in the so called Roe’s averaged condition and satisfies certain conditions which Roe termed property U . Since Ā in Eqn. 2 is locally constant, discrete expansion waves could appear as a non-physical process such as an expansion shock. The reason for the generation of this expansion shock by the Roe scheme can be better explained if the scheme is applied to the inviscid Burgers’ equation. The Burgers’ equation may be written as: ∂u ∂t + ∂ ∂x ( u 2 ) = 0 (3) 1 American Institute of Aeronautics and Astronautics Paper 2001-0083. when subjected to the Roe’s linearization method, it will take the form: ∂u ∂t + ū ∂u ∂x = 0 (4) where ū is locally constant according to Roe (1981). Therefore, the solution of Eqn. 4 in each locality is equivalent to the solution of the wave equation: ∂u ∂t + a ∂u ∂x = 0 (5) where a is the wave speed. In fact, the wave equation, Eqn. 5, exactly reproduces the initial data at a distance a · t from its original location, where t is the time elapsed. That is, the initial data without any damping, amplification or any other change has been reproduced. For the case if it was possible that the wave speed a would vanish, Eqn. 5 would reduce to ∂u/∂t = 0, for which the solution would be u =constant. That is, the initial data would be faithfully reproduced in its original location for any other time. The linearized Burgers’ equation, Eqn. 4, behaves exactly like the wave equation because ū is locally constant. For the case that ū vanishes, the initial data regardless of its configuration whether it is a compression or expansion, is faithfully reproduced. In other words, the linearized Burgers’ equation, Eqn. 4, is blind to distinguish between a compression or expansion, since each is a valid solution of Eqn. 4. However an expansion shock is not a valid solution of the original Burgers’ equation, Eqn. 3, and the correct physical solution is that the initial expansion condition must totally diffuse to a centered fan around the point of expansion. The existence of expansion shocks is a violation of the second law of thermodynamics, since it implies entropy reduction. This is corrected as shown in the following. Entropy Correction Later in this paper, it is shown that the Roe scheme is absolutely non-diffusive for grid aligned flows. For the oblique grids the amount of diffusion is very low. Therefore, they are very suitable for the computation of viscous flows, by which the viscous regions could be computed with minimum exaggeration in shear layers, e.g. boundary layers. However, in the computations of inviscid flows some non-physical solutions such as expansion shocks may occur [Hirsch (1990), Tannehill et. al. (1997), and Hoffmann and Chiang (1993)]. The non-physical expansion shocks only occur in those regions of the computational domains that expansions are observed through sonic regions, i.e. sonic expansion, in the case of Burgers’ equation in the vicinity of ū=0. In order to cure this problem we need: (1)the location of the sonic expansion in the domain of computation to be detected, and (2)the expansion shock to be avoided by diffusing the expansion shocks into expansion fans in the region of sonic expansion. Procedures (1) and (2) together are referred to as entropy correction or entropy fix. (1)Sonic expansion corresponds to the regions where the wave speed vanishes. As mentioned earlier, for the Burgers’ equation, it happens in the locality of ū ≈ 0. For the one-dimensional flow, e.g. shock tube problem, the governing equations of fluid motion could be decomposed into an equivalent system with three scalar equations each being similar to the wave equation, i.e. ∂w1 ∂t + λ1 ∂w1 ∂x = 0, ∂w2 ∂t + λ2 ∂w2 ∂x = 0, ∂w3 ∂t + λ3 ∂w3 ∂x = 0, (6) where w1, w2, and w3 are Riemann invariants, which are constant along characteristic lines dx/dt = λ1, λ2 or λ3 and λ1 = u− c, λ2 = u, λ3 = u+ c are the eigenvalues of the flux Jacobian matrix or the wave speed. For the one-dimensional flow, the regions of sonic expansion could be detected by searching the regions that |λ| approaches zero, or |λ| < , where is a small and positive number which is carefully determined. In fact is the size of the band over which the entropy correction is enforced. (2)Once the region of sonic expansion is detected, an expansion shock can be avoided by diffusing the expansion shock into the domain of computation within the band . The diffusion process is accomplished numerically by moving λ away from its origin. Various formulations could diffuse the expansion shock. The most popular is due to Harten and Hyman (1983), which maps λ to: λnew ← λ + 2 2 . (7) For example, for = 0.5 and λ = 0, the mapping of Eqn. 7 moves λ from its origin to λnew = 0.25. Some of the popularly used entropy correction formulations are given next. Accuracy Assessment of Entropy Correction Formulae In the following entropy correction formulations it should be noted that, λ̂ is the eigenvalue determined at the Roe’s averaged condition, e.g. λ̂1 = û − ĉ. Also λ = u − c and λ = u − c are the inner (L) and outer (R) value of the eigenvalue, respectively. 2 American Institute of Aeronautics and Astronautics Paper 2001-0083. 1. Harten and Hyman (1983)